Poker Nitty
poker nitty

Can anyone calculate what the chances of this occurring are?
I was playing Facebook poker(No limit Hold em') and one of the hands(starting hand, the two hole cards) that came up for me was a double King.I naturally made the most out of it, but at the end of the hand discovered that the other two guy's had got a double Seven and a double Ace.There were only three people at the table at the time and the community cards were nothing special for any of us.Obviously the guy with two Aces won.
I don't know the nitty-gritty of poker hands and probabilities, so I'm hoping one of you could tell me what the chances are of three people getting cards like that dealt out to them in the same play?
For the other readers, I'd love to hear your highly improbable poker hands as well, if you ever got any.
Three people, and each gets two hole cards, so there are
52C2 * 50C2 * 48C2 / 3!
= 1,326 * 1,225 * 1,128 / 6 = 305,377,800
different sets of hole cards dealt out to three players.
That assumes that we don't care which player gets which set in any of these combinations. Obviously, the players do care, but we're calculating the probability that the situation arises without respect to which player gets which set of hole cards.
There are 4 kings and 4 aces, so there are
4C2 = 6 different pairs of either one that a player might hold.
Generalizing the situation slightly, there are
11 * 6 = 66 different pairs that a player might hold which are lower than the king and ace pairs. (It wouldn't make much difference to the play or the outcome if the player who didn't have aces or kings had a pair other than sevens, so I am assuming any pair from deuces to queens will fit the scenario.)
That makes 6 * 6 * 66 = 2,376 different ways the three sets of hole cards could be a pair of aces, a pair of kings, and a lower pair. So the probability is
2,376 / 305,377,800 = 99 / 12,724,075 = about 0.00078% of the hands with three players that would have such a combination of hole cards.
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